We examine the way in which empirical bandwidth choice affects
distributional properties of nonparametric density estimators. Two bandwidth
selection methods are considered in detail: local and global plug-in rules.
Particular attention is focussed on whether the accuracy of distributional
bootstrap approximations is appreciably influenced by using the resample
version $\hat{h}*$,rather than the sample version $\hat{h}$, of an empirical
bandwidth. It is shown theoretically that,in marked contrast to similar
problems in more familiar settings, no general first-order theoretical
improvement can be expected when using the resampling version. In the case of
local plug-in rules, the inability of the bootstrap to accurately reflect
biases of the components used to construct the bandwidth selector means that
the bootstrap distribution of $\hat{h}*$ is unable to capture some of the main
properties of the distribution of $\hat{h}$. If the second derivative component
is slightly undersmoothed then some improvements are possible through using
$\hat{h}*$, but they would be difficult to achieve in practice. On the other
hand, for global plug-in methods, both $\hat{h}$ and $\hat{h}*$ are such good
approximations to an optimal, deterministic bandwidth that the variations of
either can be largely ignored, at least at a first-order level.Thus, for quite
different reasons in the two cases, the computational burden of varying an
empirical bandwidth across resamples is difficult to justify.